/* SPDX-License-Identifier: SunMicrosystems */
/* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. */

/**
 *
 * This family of functions implements the exponential function minus
 * :math:`1`, that is :math:`e` powered by :math:`x` before subtracting
 * :math:`1`.
 *
 * Synopsis
 * ========
 *
 * .. code-block:: c
 *
 *     #include <math.h>
 *     float expm1f(float x);
 *     double expm1(double x);
 *     long double expm1l(long double x);
 *
 * Description
 * ===========
 *
 * ``expm1`` computes :math:`e` powered by the input value subtracted by
 * :math:`1`.
 *
 * Mathematical Function
 * =====================
 *
 * .. math::
 *
 *    expm1(x) \approx e^x - 1
 *
 * Returns
 * =======
 *
 * ``expm1`` returns :math:`e` powered by :math:`x` subtracted by :math:`1`, in
 * the range :math:`\mathbb{F}_{>=-1}`.
 *
 * Exceptions
 * ==========
 *
 * Raise ``overflow`` exception when the magnitude of the input value is too
 * large.
 *
 * .. May raise ``underflow`` exception.
 *
 * Output map
 * ==========
 *
 * +---------------------+---------------+---------------+---------------+---------------+---------------+---------------+---------------+
 * | **x**               | :math:`-Inf`  | :math:`<0`    | :math:`-0`    | :math:`+0`    | :math:`>0`    | :math:`+Inf`  | :math:`NaN`   |
 * +=====================+===============+===============+===============+===============+===============+===============+===============+
 * | **expm1(x)**        | :math:`-1`    | :math:`e^x-1` | :math:`+0`                    | :math:`e^x-1` | :math:`+Inf`  | :math:`qNaN`  |
 * +---------------------+---------------+---------------+---------------+---------------+---------------+---------------+---------------+
 *
 *///

/* expm1(x)
 * Returns exp(x)-1, the exponential of x minus 1.
 *
 * Method
 *   1. Argument reduction:
 *    Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
 *
 *      Here a correction term c will be computed to compensate
 *    the error in r when rounded to a floating-point number.
 *
 *   2. Approximating expm1(r) by a special rational function on
 *    the interval [0,0.34658]:
 *    Since
 *        r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
 *    we define R1(r*r) by
 *        r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
 *    That is,
 *        R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
 *             = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
 *             = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
 *      We use a special Remez algorithm on [0,0.347] to generate
 *     a polynomial of degree 5 in r*r to approximate R1. The
 *    maximum error of this polynomial approximation is bounded
 *    by 2**-61. In other words,
 *        R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
 *    where     Q1  =  -1.6666666666666567384E-2,
 *              Q2  =   3.9682539681370365873E-4,
 *              Q3  =  -9.9206344733435987357E-6,
 *              Q4  =   2.5051361420808517002E-7,
 *              Q5  =  -6.2843505682382617102E-9;
 *      (where z=r*r, and the values of Q1 to Q5 are listed below)
 *    with error bounded by
 *        |                  5           |     -61
 *        | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
 *        |                              |
 *
 *    expm1(r) = exp(r)-1 is then computed by the following
 *     specific way which minimize the accumulation rounding error:
 *                   2     3
 *                  r     r    [ 3 - (R1 + R1*r/2)  ]
 *          expm1(r) = r + --- + --- * [--------------------]
 *                      2     2    [ 6 - r*(3 - R1*r/2) ]
 *
 *    To compensate the error in the argument reduction, we use
 *        expm1(r+c) = expm1(r) + c + expm1(r)*c
 *               ~ expm1(r) + c + r*c
 *    Thus c+r*c will be added in as the correction terms for
 *    expm1(r+c). Now rearrange the term to avoid optimization
 *     screw up:
 *                (      2                                    2 )
 *                ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
 *     expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
 *                    ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
 *                      (                                             )
 *
 *           = r - E
 *   3. Scale back to obtain expm1(x):
 *    From step 1, we have
 *       expm1(x) = either 2^k*[expm1(r)+1] - 1
 *            = or     2^k*[expm1(r) + (1-2^-k)]
 *   4. Implementation notes:
 *    (A). To save one multiplication, we scale the coefficient Qi
 *         to Qi*2^i, and replace z by (x^2)/2.
 *    (B). To achieve maximum accuracy, we compute expm1(x) by
 *      (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
 *      (ii)  if k=0, return r-E
 *      (iii) if k=-1, return 0.5*(r-E)-0.5
 *        (iv)    if k=1 if r < -0.25, return 2*((r+0.5)- E)
 *                      else         return  1.0+2.0*(r-E);
 *      (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
 *      (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
 *      (vii) return 2^k(1-((E+2^-k)-r))
 *
 * Special cases:
 *    expm1(INF) is INF, expm1(NaN) is NaN;
 *    expm1(-INF) is -1, and
 *    for finite argument, only expm1(0)=0 is exact.
 *
 * Accuracy:
 *    according to an error analysis, the error is always less than
 *    1 ulp (unit in the last place).
 *
 * Misc. info.
 *    For IEEE double
 *        if x >  7.09782712893383973096e+02 then expm1(x) overflow
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

#include <math.h>
#include "../common/tools.h"

#ifndef __LIBMCS_DOUBLE_IS_32BITS

static const double
one         =  1.0,
o_threshold =  7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
ln2_hi      =  6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
ln2_lo      =  1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
invln2      =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
/* scaled coefficients related to expm1 */
Q1          = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
Q2          =  1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
Q3          = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
Q4          =  4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
Q5          = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */

double expm1(double x)
{
#ifdef __LIBMCS_FPU_DAZ
    x *= __volatile_one;
#endif /* defined(__LIBMCS_FPU_DAZ) */

    double y, hi, lo, c, t, e, hxs, hfx, r1;
    int32_t k, xsb;
    uint32_t hx;

    GET_HIGH_WORD(hx, x);
    xsb = hx & 0x80000000U;      /* sign bit of x */

    hx &= 0x7fffffff;        /* high word of |x| */

    /* filter out huge and non-finite argument */
    if (hx >= 0x4043687A) {           /* if |x|>=56*ln2 */
        if (hx >= 0x40862E42) {       /* if |x|>=709.78... */
            if (hx >= 0x7ff00000) {
                uint32_t low;
                GET_LOW_WORD(low, x);

                if (((hx & 0xfffff) | low) != 0) {
                    return x + x;    /* NaN */
                } else { /* exp(+-inf)={inf,-1} */
                    return (xsb == 0) ? x : -1.0;
                }
            }

            if (x > o_threshold) {
                return __raise_overflow(one);    /* overflow */
            }
        }

        if (xsb != 0) { /* x < -56*ln2, return -1.0 with inexact */
            return -__raise_inexact(one);    /* return -1 */
        }
    }

    /* argument reduction */
    if (hx > 0x3fd62e42) {       /* if  |x| > 0.5 ln2 */
        if (hx < 0x3FF0A2B2) {   /* and |x| < 1.5 ln2 */
            if (xsb == 0) {
                hi = x - ln2_hi;
                lo =  ln2_lo;
                k =  1;
            } else {
                hi = x + ln2_hi;
                lo = -ln2_lo;
                k = -1;
            }
        } else {
            k  = invln2 * x + ((xsb == 0) ? 0.5 : -0.5);
            t  = k;
            hi = x - t * ln2_hi;  /* t*ln2_hi is exact here */
            lo = t * ln2_lo;
        }

        x  = hi - lo;
        c  = (hi - x) - lo;
    } else if (hx < 0x3c900000) {   /* when |x|<2**-54, return x */
        if (x == 0.0) {
            return x;
        } else { /* return x with inexact flags when x!=0 */
            return __raise_inexact(x);
        }
    } else {
        k = 0;
    }

    /* x is now in primary range */
    hfx = 0.5 * x;
    hxs = x * hfx;
    r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
    t  = 3.0 - r1 * hfx;
    e  = hxs * ((r1 - t) / (6.0 - x * t));

    if (k == 0) {
        return x - (x * e - hxs);    /* c is 0 */
    } else {
        e  = (x * (e - c) - c);
        e -= hxs;

        if (k == -1) {
            return 0.5 * (x - e) - 0.5;
        }

        if (k == 1) {
            if (x < -0.25) {
                return -2.0 * (e - (x + 0.5));
            } else {
                return  one + 2.0 * (x - e);
            }
        }

        if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
            uint32_t high;
            y = one - (e - x);
            GET_HIGH_WORD(high, y);
            SET_HIGH_WORD(y, high + (((uint32_t)k) << 20)); /* add k to y's exponent */
            return y - one;
        }

        t = one;

        if (k < 20) {
            uint32_t high;
            SET_HIGH_WORD(t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
            y = t - (e - x);
            GET_HIGH_WORD(high, y);
            SET_HIGH_WORD(y, high + (k << 20)); /* add k to y's exponent */
        } else {
            uint32_t high;
            SET_HIGH_WORD(t, ((0x3ff - k) << 20)); /* 2^-k */
            y = x - (e + t);
            y += one;
            GET_HIGH_WORD(high, y);
            SET_HIGH_WORD(y, high + (k << 20)); /* add k to y's exponent */
        }
    }

    return y;
}

#ifdef __LIBMCS_LONG_DOUBLE_IS_64BITS

long double expm1l(long double x)
{
    return (long double) expm1((double) x);
}

#endif /* #ifdef __LIBMCS_LONG_DOUBLE_IS_64BITS */
#endif /* #ifndef __LIBMCS_DOUBLE_IS_32BITS */
